All the above rely on the concept of phason engineering. Using this strategy, the Hall physics from an arbitrarily high dimension can be access from a physical space of lower dimension \(d=1,2,3\) . The high throughput of large classes of topological models produced by phason engineering will be essential for our understanding of higher-dimensional bulk-boundary correspondence, of its manifestation in meta-materials and of its possible applications. Phason engineering relies on a specialized algorithm to position the resonators relative to each other to produce phasons that live on arbitrary \(d^{\prime }\) -dimensional tori. Regardless of the particular couplings of the resonators, the dynamical matrices that determine the collective resonant spectrum of the crystal are Galilean invariant. We show that any such Galilean invariant dynamical matrix is just linear combinations and products of elementary operators that satisfy the commutation relations of the magnetic translations in \((d+d^{\prime })\) -dimensions. As such, the spectral gaps of the crystals carry higher Chern numbers and they display a bulk-boundary correspondence specific to IQHE in higher dimensions.[1]} Furthermore, to navigate the complex topology of the states, we devise a K-theoretic visual method to map the large number of topological invariants associated with the bulk gaps, based on the gap labeling technique.[2]}, [3]}.
